Distance from the source

\(D\) is calculated with a least-cost pathfinding algorithm from the R package gdistance, by establishing a connectivity model based on a ‘resistance seascape’ concept. We created a 100 x 100 m raster whose cell can be selected to obtain the path connecting start (source of the activity) and end (each raster cell) points. Journey between two neighbour cells has a cost to be included in this path, which is dependant on several constraints. The length of the final path then gives distance \(D_{ij}\).

We considered three underlying principles for the physical constraints:

• marine ecosystems: particles cannot disperse on land
• gravity: particles disperse easily from shallow to deeper depths while the reverse is difficult
• hydrodynamism: particles disperse according to local hydrodynamical currents

The connectivity model thus included: (i) coasts as boundaries delimitating cells unselectable by the algorithm, (ii) bathymetry, (iii) river plumes as hydrodynamical fronts with an intensity and a direction (a complete circulation model in BSI is not yet available). These constraints were implemented in the transition function \(t\), used by costDistance() when creating the least-cost path (chess queen configuration):

• land is set with a connectivity of 0
• bathymetry is compared between two cells:
  • when point A is shallower than point B, connectivity is highest
  • when point A is deeper than point B, connectivity is lower
• CDOM content (proxy of hydrodynamism) is compared between two cells

The equation for \(D_{ij}\) is:

\[ D_{ij} = t(S_{j}, G_{i}, B_{i}, H_{i}) \]

• \(t\) is the transition function
• \(S_{j}\) is the source(s) of activity
• \(G_{i}\) is the geography component (coastline)
• \(B_{i}\) is the bathymetry component
• \(H_{i}\) is the hydrodynamics component (CDOM proxy)
• \(i\) is a cell
• \(j\) is a human activity

⚠️ For now, the hydrodynamics constraint is not implemented.

Exposure

With \(D_{ij}\) calculated, we can compute \(E_{ij}\) with the decay function. To this end, we used a gaussian kernel function (exponential quadratic relationship), which is a way to consider diffusion in a 2D environment:

\[ E_{ij} = exp\left(C . a_{j} . (D_{ij} - b_{j})^2 + c_{j} \right) \]

• \(C\) is a constant
• \(a_{j}\), \(b_{j}\), \(c_{j}\) are parameters of the decay function
• \(D_{ij}\) is the distance from the source
• \(i\) is a cell
• \(j\) is a human activity

The constant \(C\) is set to 0.000025, linked to the spatial extent of the study area (\(range\) = 25000 m). The general equation to set \(C\) would be:

\[ C = \frac{1}{range} \]

We considered five decay functions to model different diffusion behaviours for human activities. To do so, we modified parameters of this function, in particular \(a_{j}\) (\(b_{j}\) and \(c_{j}\) are set to 0):

• Type I: very localized ( purple curve, \(a_{j}\) = -1)
• Type II: localized ( blue curve, \(a_{j}\) = -0.1)
• Type III: diffused ( green curve, \(a_{j}\) = -0.01)
• Type IV: very diffused ( yellow curve, \(a_{j}\) = -0.001)
• Type V: ubiquitous ( red curve, \(a_{j}\) = -0.0001)

To assign a type of decay function to a human activity, we performed a litterature review describing the drivers related to each considered human activity and how can we model them best (details will be available in the final version of the article). Based on this work, here are the results: